In the limit \(\omega \rightarrow 0\)  the classical action Eq. (REF ) reduces to the well-known formula[1]}, [2]}:
\({S}_{\text{c}}^{\text{B}}=\frac{m\omega _{L}}{2\tan {{\omega _{L}T}}}\left[(x_b-x_a)^2 + (y_b-y_a)^2\right]-m\omega _{L}(y_a x_b-x_a y_b)\) 
